Complex Geometry

Course Overview

This graduate-level course offers an introduction to the fundamental concepts and techniques of complex differential geometry.

The central aim of the course is to understand the criteria that determine when a compact complex manifold can be realized as a smooth projective algebraic variety. This is the celebrated Kodaira Embedding Theorem, a cornerstone result that provides a precise differential-geometric condition (the existence of a positive line bundle, or a Hodge metric) for a complex manifold to be projective (and thus algebraic by Chow's theorem). We will work through the necessary machinery to fully prove this theorem.

Time permitting, we will then discuss Kodaira-Spencer deformation theory and discuss the case of Calabi-Yau manifolds, studied by Tian-Todorov.

Prerequisites: Complex analysis and differential geometry. Familiarity with Riemannian geometry and vector bundles is desirable.

Notes and references

I will draw from

• D. Huybrechts, Complex Geometry: An Introduction


• J.-P. Demailly, Complex Analytic and Differential Geometry

I will (try to) keep up-to-date lecture notes as the term progresses, available here (Last update: 22^ndof Septmeber,2025)

If you find any mistakes or typos, please let me know by dropping me an email.

Syllabus

0. Overview


1. Holomorphic functions


2. Complex and almost complex manifolds


3. Sheaves and their cohomologies


4. Holomorphic bundles, Kodaira dimension and Siegel's theorem


5. Divisors and blow-ups


6. Metrics and connections


7. The Kähler condition


8. Positivity and vanishing


9. The Kodaira embedding theorem


10. Kodaira-Spencer deformation theory


11. (Formal) Tian-Todorov theorem

Exercises

There will be a list of exercises published weekly. You are encouraged to work through them and discuss them with your colleagues No official soluion will be provided.

• Week 1
• Week 2
• Week 3